Theorem dnnumch2 43894

Description: Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
dnnumch.f	⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
dnnumch.g	⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))

Assertion

Ref	Expression
dnnumch2	⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

Distinct variable groups:   𝑦,𝐹   𝑦,𝐺,𝑧   𝑦,𝐴,𝑧

Allowed substitution hints:   𝜑(𝑦, 𝑧)   𝐹(𝑧)   𝑉(𝑦, 𝑧)

Proof of Theorem dnnumch2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dnnumch.f	. . 3 ⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2		dnnumch.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		dnnumch.g	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))
4	1, 2, 3	dnnumch1 43893	. 2 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
5		f1ofo 6829	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → (𝐹 ↾ 𝑥):𝑥–onto→𝐴)
6		forn 6796	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
7	5, 6	syl 18	. . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
8		resss 5998	. . . . . 6 ⊢ (𝐹 ↾ 𝑥) ⊆ 𝐹
9		rnss 5927	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥) ⊆ 𝐹 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹)
10	8, 9	mp1i 14	. . . . 5 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → ran (𝐹 ↾ 𝑥) ⊆ ran 𝐹)
11	7, 10	eqsstrrd 3969	. . . 4 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹)
12	11	a1i 11	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹))
13	12	rexlimdvw 3170	. 2 ⊢ (𝜑 → (∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 → 𝐴 ⊆ ran 𝐹))
14	4, 13	mpd 16	1 ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560   ↦ cmpt 5190  ran crn 5660   ↾ cres 5661  Oncon0 6361  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  recscrecs 8363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364

This theorem is used by:  dnnumch3lem  43895  dnnumch3  43896